Question

At what temperature are Fahrenheit and Celsius temperatures the same in value but opposite in sign?

Short answer

At \(C = -11.43\) degrees, the temperature in Fahrenheit is \(11.43\) degrees.

Step-by-step solution

Understand the Relationship between Fahrenheit and Celsius

The formula to convert Celsius (C) to Fahrenheit (F) is given by \[ F = \frac{9}{5}C + 32 \] We need to find a temperature at which these temperatures are numerically equal but opposite in sign, i.e., \[ F = -C \]

Set up the Equation

Substitute \(F = -C\) into the conversion formula: \[ -C = \frac{9}{5}C + 32 \]

Solve for Celsius

Rearrange the equation to isolate C: \[ -C - \frac{9}{5}C = 32 \] Combine terms on the left side: \[ -\frac{5}{5}C - \frac{9}{5}C = 32 \] \[ -\frac{14}{5}C = 32 \] Solve for C: \[ C = -\frac{32 \times 5}{14} \] \[ C = -\frac{160}{14} \] \[ C = -\frac{80}{7} \] \[ C ≈ -11.43 \]

Verify the Fahrenheit Temperature

Check if \(F = 11.43\) when \(C = -11.43\): Substitute \(C = -11.43\) into the conversion formula: \[ F = \frac{9}{5}(-11.43) + 32 \] \[ F ≈ -20.574 + 32 \] \[ F ≈ 11.43 \] This confirms our solution.

Key concepts

Fahrenheit to Celsius conversion

To understand how temperatures are converted between Fahrenheit and Celsius, you need to know the formula that relates these two scales. This conversion formula is:

• Fahrenheit (F) = \[ \frac{9}{5}C + 32 \]
• Celsius (C) = \[ \frac{5}{9}(F - 32) \]

When you look at this formula, notice that multiplying a Celsius temperature by \[ \frac{9}{5} \] and adding 32 converts it to Fahrenheit. This is because Celsius and Fahrenheit scales are designed with different zero points and increments. Understanding this relationship is essential since it helps achieve accurate temperature conversions.

solving equations

Solving equations involves finding the value of the unknown that makes the equation true. In this exercise, we were asked to find a specific condition where Fahrenheit and Celsius temperatures are equivalent but have opposite signs. First, we substituted \[ F = -C \] into our conversion formula, resulting in \[ -C = \frac{9}{5}C + 32 \].Next, we rearranged the equation to combine like terms. This led us to solve for \[ C \text{ (the Celsius temperature)} \]:

• Combine terms: \[ -C - \frac{9}{5}C = 32 \].
• Simplify: \[ -\frac{14}{5}C = 32 \].
• Solve for C: \[ C = -\frac{160}{14} \] or approximately \[ -11.43°C \].

Lastly, we verified our solution by calculating the corresponding Fahrenheit temperature and checking if it matched our expectation. Correct verification confirmed the accuracy of our solution.

temperature scales

Temperature scales, such as Fahrenheit and Celsius, are methods of measuring temperature where each scale has specific reference points.

• In the Celsius scale, 0°C is the freezing point of water, and 100°C is the boiling point.
• In the Fahrenheit scale, 32°F is the freezing point of water, and 212°F is the boiling point.

These scales also differ in increments and zero points. Converting between these scales requires understanding these differences and using specific conversion formulas. This is why we derived an exact point (-11.43°C or approximately 11.43°F) where Celsius and Fahrenheit values are numerically equal but opposite in sign.